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arXiv · 1907.10474

On the Cheeger problem for rotationally invariant domains

Vladimir Bobkov·Enea Parini
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Abstract

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $\Omega \subset \mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $\partial C \cap \Omega$ consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show that if $\Omega$ is convex, then the free boundary of $C$ consists only of pieces of spheres and nodoids. This result remains valid for nonconvex domains when the generating curve of $C$ is closed, convex, and of class $\mathcal{C}^{1,1}$. Moreover, we provide numerical evidence of the fact that, for general nonconvex domains, pieces of unduloids or cylinders can also appear in the free boundary of $C$.

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Vladimir Bobkov, Enea Parini. 2019-07-24. On the Cheeger problem for rotationally invariant domains. https://doi.org/10.1007/s00229-020-01260-9

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